Trenitalia's high speed and long distance trains tickets are never sold 2 or 3 months in advance, it's 6 months in advance or more. I'm afraid the above info is wrong, whereas it may be right regarding discounted tickets.
I'll try to explain how dynamic pricing works. From Genoa to Turin Trenitalia runs 6 direct Intercity trains, but only one direct high speed train: the Frecciarossa #8606 that departs from Genoa Piazza Principe station at 11:56 AM.
The Freccia #8606 is the most high-priced option between Genoa and Turin. If you enter December 12 as day of travel, only the expensive BASE (aka walk-up) fare tickets are available. But if you enter August 31, the system shows many discounted tickets on sale for the same 8606 train. BASE: 33 € per person vs SuperEconomy: 14.9 €.
Between Milan and Genoa Trenitalia runs 2 direct Freccias and 14 direct Intercity trains any day. It works the same way: in short, If you see only BASE tickets it's either too late or too early.
The Intercity trains departing from Genoa are a little slower and a little cheaper than the Freccias, simply because there are no high speed tracks running under the mountains north of Genoa. Not yet, but they keep on digging.
Of course you could also take the slow and cheap Regionale trains that depart daily from Genoa. The Regionale trains make many intermediate stops on the way to Turin and Milan, but they have no reserved seating and can't sell out. The price of Regionale trains is fixed, no discounts for early birds.
beware the requests for you to OK 'added guaranteed coverage' or somesuch BS; we just got charged 12 euros for that even though we never asked for, nor OKed it.
The Ticket Refund option for discounted fares costs 2 Euro per ticket, not 12. You must accept the T&C at least twice and it's a great option when you get tickets so early.